Anomalous fractional quantization in the kagomelike Heisenberg ladder: Emergence of the effective spin-1 chain

We study a kagomelike spin-$\frac{1}{2}$ Heisenberg ladder with competing ferromagnetic (FM) and antiferromagnetic (AFM) exchange interactions. Using the density-matrix renormalization group based calculations, we obtain the ground-state phase diagram as a function of the ratio between the FM and AFM exchange interactions. Five different phases exist. Three of them are spin-polarized phases: an FM phase and two kinds of ferrimagnetic (FR) phases (referred to as FR1 and FR2 phases). The spontaneous magnetization per site is $m=\frac{1}{2}, \frac{1}{3}$, and $\frac{1}{6}$ in the FM, FR1, and FR2 phases, respectively. This can be understood from the fact that an effective spin-1 Heisenberg chain formed by the upper and lower leg spins has a three-step fractional quantization of the magnetization per site as $m=1, \frac{1}{2}$, and 0. In particular, an anomalous “intermediate” state $m=\frac{1}{2}$ of the effective spin-1 chain with the reduced Hilbert space of a spin from three to two dimensional is highly unexpected in the context of conventional spin-1 physics. Thus, surprisingly, the effective spin-1 chain behaves like a spin-$\frac{1}{2}$ chain with SU(2) symmetry. The remaining two phases are spin-singlet phases with translational symmetry breaking in the presence of valence bond formations. One of them is octamer-singlet phase with a spontaneous octamerization long-range order of the system, and the other is period-4 phase characterized by the magnetic superstructure with a period of four structural unit cells. In these spin-singlet phases, we find the coexistence of valence bond structure and gapless chain. Although this may be emerged through the order-by-disorder mechanism, there can be few examples of such a coexistence.

Figure 1

(a), (b) Lattice structure of the kagomelike Heisenberg ladder with the exchange interactions $J_1, J_2$, and $J_3$, where ${\mathit{\sigma }}_i$ and ${\mathit{\xi }}_i$ are called the upper and lower leg spins, respectively, and ${\mathit{s}}_i$ is called the axial (or central) spin. The kagomelike Heisenberg ladder consists of corner-sharing triangles created by wrapping the kagome lattice to form a cylinder. (c) Schematic ground-state phase diagram of our model as a function of $J_b=J_2=J_3$ (see the main text). The abbreviations FM, FR, OS, and P4 denote the ferromagnetic, ferrimagnetic, octamer-singlet, and period-4 phases, respectively.

Figure 2

Finite-size scaling analysis of $S_{\mathrm{tot}}/L$ calculated using the (a) OBC and (b) PBC, where representative parameters for each of the FM (red circles), FR1 (blue squares), OS (green triangles), FR2 (purple diamonds), and P4 (cyan stars) phases are chosen. (c) iDMRG results for $\langle S_{\alpha ,i}^z\rangle$ as a function of $J_b/\left|J_1\right|$: $\langle S_s^z\rangle$ for the axial spin and $\langle S_{\sigma }^z\rangle =\langle S_{\xi }^z\rangle$ for the leg spins are shown by red circles and blue squares, respectively. The $z$ component of total spin is fixed at the value of total spin in the ground state. The averaged magnetization per site $m=(\langle S_s^z\rangle +\langle S_{\sigma }^z\rangle +\langle S_{\xi }^z\rangle )/3$ is also shown by black diamonds. Horizontal dotted lines indicate the magnetization per site for the FR1 ($m=\frac{1}{3}$) and FR2 ($m=\frac{1}{6}$) phases calculated. The regions of different phases are divided by different colors.

Figure 3

DMRG results for the static spin structure factor $S_{\alpha }\left(q\right)$ calculated using $L=72$ PBC cluster. (a)–(d) $S_{\alpha }\left(q\right)$ with typical values of $J_b/\left|J_1\right|$ for each phase: FR1 ($J_b/\left|J_1\right|=0.3$), OS ($J_b/\left|J_1\right|=0.4$), FR2 ($J_b/\left|J_1\right|=0.5$), and P4 ($J_b/\left|J_1\right|=1.0$). Bottom panels show the contour map of $S_{\alpha }\left(q\right)$ for (e) the axial spins $\alpha =s$ and (f) the leg spins $\alpha =\sigma =\xi$ as a function of $J_b/\left|J_1\right|$. The four vertical dashed lines correspond to phase boundary in the ground state.

Figure 4

iDMRG results for the spin-spin correlation function in the real space, where red circles and blue squares correspond to $S_s\left(r\right)$ and $S_{\sigma }\left(r\right)$, respectively. Note that $S_{\xi }\left(r\right)$ is equivalent to $S_{\sigma }\left(r\right)$. $J_b/\left|J_1\right|$ values for each phase are chosen: (a), (c) FR1 ($J_b/\left|J_1\right|=0.3$), (b), (d) FR2 ($J_b/\left|J_1\right|=0.5$), (e), (f) OS ($J_b/\left|J_1\right|=0.4$), and (g), (h) P4 ($J_b/\left|J_1\right|=1.0$). In (c) and (d) only the oscillating part of the spin-spin correlation function is plotted. The dotted straight lines indicate the characteristic decays, namely, a power-law decay in the log-log plot and and exponential decay in the semi-log plot.

Figure 5

Finite-size scaling analysis of the spin gap ${\mathrm{\Delta }}_{\mathrm{s}}\left(L\right)$. The representative $J_b/\left|J_1\right|$ values for each phase are chosen: FR1 ($J_b/\left|J_1\right|=0.3$), OS ($J_b/\left|J_1\right|=0.4$), FR2 ($J_b/\left|J_1\right|=0.5$), and P4 ($J_b/\left|J_1\right|=1.0$). The lines show fitting results by polynomial functions with respect to $1/L$.

Figure 6

(a) A six-site PBC cluster of the kagomelike ladder, which may be a minimal unit to describe the FR1 state. (b) Schematic picture of the ground state of the six-site PBC cluster for $0.25<J_b/\left|J_1\right|<0.5$, which corresponds to the FR1 state with $S_{\mathrm{tot}}/L=\frac{1}{3}$. A blue ellipse denotes spin-triplet pair between an upper and a lower spin, which corresponds to an effective spin-1 degrees of freedom and a red arrow denotes a polarized spin. (c) Schematic representation of the magnetic moment distributions in real space for the FR1 phase, where the magnetization of each spin-triplet pair (denoted by blue ellipse) is either $m=0$ or 1. Note that the numbers of $m=0$ and 1 are equivalent, namely, the averaged value is $m=\frac{1}{2}$, in the ground state.

Figure 7

(a), (b) Structure of the octamer singlet. (c) iDMRG results for the spin-spin correlations as a function of $J_b/\left|J_1\right|$ around the OS phase. The symbols and colors correspond to the lattice bonds denoted in (a) and (b).

Figure 8

(a) Schematic representation of the magnetic moment distributions in real space for the FR2 phase, where the weakly polarized spins are denoted by red shaded area. A red arrow denotes an almost fully polarized spin. (b), (c) FM-AFM delta chain as an effective model for the FR2 phase, where each basal spin is $S=1$ as a result of two $S=\frac{1}{2}$ leg spins.

Figure 9

(a) Structure of effective exchange couplings $J_s^{\mathrm{NN}}$ and $J_s^{\mathrm{NNN}}$ for the axial-spin subsystem (see text). (b) iDMRG results for the spin-spin correlations as a function of $J_b/\left|J_1\right|$ in the P4 phase. The symbols and colors correspond to the lattice bonds denoted in (c). Cyan stars show the dimerization order parameter, which is estimated by Eq. (17) and is equivalent to the difference between the values denoted by red circles and blue squares.

Figure 10

(a) DMRG and iDMRG results for the $z$ component of a nonlocal string correlation function $O_{\mathrm{s},\mathrm{odd}}\left(r\right)$ as a function of distance $r$. The symbols represent OBC results with $L=300$, and lines represent the corresponding iDMRG results. The cyan sold line shows the result for $J_1=0$, where the system is decomposed into the so-called diagonal ladder and free axial spins (see main text). The other numbers inside the figure are the values of $J_b/\left|J_1\right|$. (b) Finite-size scaling analysis of the string order parameter $O_{\mathrm{s},\mathrm{odd}}(r=N/2)$ calculated using OBC clusters.

Figure 11

(a) A possible realization of octamer-singlet state with an open cluster. (b) DMRG results for the expectation value of the $z$ component of spin operator $\langle S_i^z\rangle$ in the $S^z=2$ sector, as a function of site index $i$. $J_b/\left|J_1\right|=0.4$ and $L=44$ are chosen. Red circles, blue squares, and green triangles represent $s, \sigma$, and $\xi$ site, respectively. Filled red circles denote the values of $\langle S_i^z\rangle$ for nearly free spins at $i=9n$, where $n$ is an integer.

Figure 12

Excitation gap, defined as an energy difference between the $S^z=0$ ground state and $S^z=1$ excited state, of isolated octamer. The red shaded area represents the octamer-singlet phase of the kagomelike chain.

Figure 13

DMRG results of static structure factor $S_{ss}\left(q\right)$ for the axial spins in the kagomelike chain at $J_b/\left|J_1\right|=0.8$, where the open boundary conditions are used. The maximum position is marked by blue diamond and the corresponding $q$ value is shown as $q_{\max }$.